Summary: HELMER ASLAKSEN
RESTRICTED HOMOGENEOUS COORDINATES FOR
THE CAYLEY PROJECTIVE PLANE
ABSTRACT.I. Porteous has shownthat the Cayleyprojectiveplanecan becoordinatizedin a way
resembling homogeneous coordinates. We will show how to construct line coordinates in a
similarway.As an illustration, we givean explicitexampleto showthat the Cayleyprojective
plane is not Desarguean.
1. INTRODUCTION
It is well known that since the Cayley numbers are not associative, we cannot
coordinatize the Cayley projective plane by homogeneous coordinates.
Instead we can either use inhomogeneous coordinates as described in, for
instance, [Ha] or represent the points as 3 × 3 Cayley valued Hermitian
matrices as introduced by Jordan [Jo] and Freudenthal [Fr]. Porteous [Po]
has shown, however, that we can use the fact that the Cayley numbers form
an alternative division algebra to introduce a type of coordinates that we will
call restricted homogeneous coordinates. Porteous only discusses point coor-
dinates, and it is not immediately clear how to define line coordinates in a
way that does not require associativity. The purpose of this brief article is to
show how restricted homogeneous coordinates can be used to describe the
lines in the Cayley projective plane. As an example of how to compute with